Trade and the Development of Numerical Systems
Trading Numbers
Trading Numbers
Vasco da Gama was among the first European explorers to come in contact with a hunter-gatherer society in southern Africa later dubbed the ‘Hottentots’, a name disparagingly intended to impersonate the sound the group’s native language (Schapera, 1930). This tribe, now known as the Khoi (or Khoikhoi), was initially viewed by Europeans as the ‘‘outcasts of the human species, a race whose intellectual faculties are so little superior to those of beasts, that some have been inclined to suppose them more nearly related to baboons than to men’ (Banks, 1896). These words, penned by Sir Joseph Banks at the end of the 19th century, typified the beliefs of “worldly” Europeans at the time. Their views became even more entrenched when descriptions of the Khoi’s numerical system reached Europe soon thereafter (Symons, 2006), indicating the extent of their number words to be limited to “one”, “two”, and “many.”. Early behavioral research had indicated that some animals possess numerical competency up to about three or four before their competence falls off. This ability called “subitizing” has been described more generally as the “rapid identification and labeling of small simultaneously presented items” (Davis & Perussé 1988), and seems to be innate in intelligent animals.
Although their intention and conclusions are very different, contemporary researchers have established that the Khoi language is limited to a one-to-many numerical space, such that a Khoi man who has four goats would report the same number word to quantify his herd as a woman with a herd of three goats. Recently, the story has become more complex with the realization that these number words are not of set value and may correspond better to Westerners’ conception of “amount”, at least in the case of the numerical system of the Piraha South American, another illiterate, nomadic tribe (see Figure 1, Frank et al., 2008). Although the Piraha can learn to use larger numbers, they do not teach basic mathematical skills like counting to their young, and thereby have more difficulty with tasks involving memory of amounts greater than 3 or 4 than those familiar with counting. The Piraha’s subitizing abilities and their basic estimation skills, however, are identical to Westerners’. The results of this and other studies strongly suggest that facility with numbers and counting are cognitive skills not universal among humans, as was once assumed. Frank et al. (2008) further suggest that the numerosity concepts in the Piraha language likely resemble those of the vast majority of prehistoric languages from which modern numerals and notational systems evolved.
Embellishment of Subitizing and Estimation If all modern cardinal number systems evolved from simple one-to-many systems, then the emergence of counting and more complicated systems of writing and representing numerosity were likely created and popularized as a result of changing environmental demands. In this light, the development of (spoken) numbers and corresponding (written) numerals highlights the birth of novel, abstract human concerns, less constrained by day-to-day circumstances and more cognizant of the future. This level of sophistication corresponded with the steady accumulation of wealth accrued by sedentary agricultural societies in the Middle East and China. This, along with developments in writing, trade and government, created a piece of global intellectual property that became so universal that people lacking such constructs were deemed be primitive and inferior. By studying the development of numbers, however, one learns that the ways in which modern Westerners have come to think about numerosity are very new, evolving long after humans diverged from other apes.
A Digital Stone Age. It is no coincidence that many number-related words are rooted in words signifying physical aids commonly used in the counting of large quantities. The most ubiquitous example is the word digit that is based on the Latin word digitus, which roughly translates to ‘finger’ or ‘toe’. The common practice of using fingers or toes to keep track of quantity also gave birth to the base 10 metric system used today. Although the word digit has retained some of its original meaning, other words like calculate and calculus that were derived from the Latin word meaning “pebble”, have come to represent exclusive mathematical terms mostly detached from their etymological root. Because we know very little specific information about prehistoric language, however, it is difficult to study the evolution of the conceptualization of numbers prior to the advent of writing some 5,000 years ago (Schmandt-Besserat, 1982). Nevertheless, there is evidence of humans using visual symbols to represent numerical concepts that date back much longer, and these artifacts suggest that writing may have been partially inspired by this practice.
The first known evidence of human mathematical inquiry is an assemblage of bones dated around 35,000 B.C. with sets of 28 tally marks deliberately etched into their sides (Bogoshi et al., 1987). These artifacts, called Lebombo bones (see Figure 2), mark the beginnings of the still common practice of tracking quantity with tally marks. However, this method becomes less and less viable as the quantities under consideration become increasingly larger. Before humans became sedentary, however, having to account for very large quantities was probably rare in their “15 minute culture” (Foley & Gamble, 2009), where all of one’s possessions were carried from campsite to campsite. In such conditions, a numerical system like that employed by the Khoi and Piraha may have been much more practical and efficient. Only when humans settled down and began to amass property and wealth did the need for more expansive systems arise. Visions of Numerals. Humans are visual creatures. We have evolved to rely most heavily on visual information available in varied environments. Although other animals can similarly conceptualize small quantities, it wasn’t until humans began to represent the concept of numbers visually, that is pictorially, that they could apply more detail to it. One of the most influential early systems of numerals developed in the Indus Valley between 3,000 and 4,000 BC, probably following the invention of writing in neighboring Mesopotamian civilizations (Sarma,1973). The early Brahmi system comprised individual symbols for the numbers 1 through 9, multiples of 10 from 10-90, multiples of 100 from 100-1000 and so forth. This base 10 system operated in much the same way Roman numerals do, as it lacked a symbol for zero. As shown in Figure 3, many of these symbols show an uncanny resemblance to the system of numerals used in this very text.
Counting on Position. Around 2,000 BC in what is present-day Iraq, the ancient Sumerians added “positional enumeration”, a powerful alteration to systems akin to the Brahmi numerals. If early attempts to symbolize numbers took advantage of humans’ acute visual abilities, then the Sumerians took advantage of language, another human anomaly. Prominent linguists like Noam Chomsky have described recursion as a universal property of human languages. Recursion refers to the capacity of language to represent an infinite number of symbols with a finite number of discrete symbols by combining them in highly specified ways. Although the Sumerian system was base 60 and lacked a symbol for zero, it was perhaps the first written numerical system to utilize recursion. Unlike previous systems, which required the invention of new symbols to represent large quantities, the Sumerian system could construct an infinite set of numbers using just the 60 basic symbols shown in Figure 3. While the ability to interpret positional systems may employ otherwise linguistic cognitive functions, that is not to say that spoken representations of numbers retained the efficiency granted by positional enumeration. The basic counting scheme taught to all Western children, for instance, operates with about the efficiency of the Brahmi system when describing large values (e.g., you have to say “six-billion, six-hundred-ninety-two-million, thirty-thousand, two-hundred-seventy-seven” to report the world population succinctly written as 6,692,030,277). The Number that Never Existed. Because the ancient Sumerians had 60 basic digits in their numeric system, the need to use zero as a positional place holder was probably rare. Since they seemed unsure whether zero was a legitimate digit, early writers either completely omitted any reference to it, or entered a numerical place holder where the zero should fall. This all changed in the Persian era (around 300 BC) when zero was first represented with its own symbol of two slanted wedges (Boyer, 1944). The conception and symbolization of zero was also achieved by the Mayans about a millennium later. As shown in Figure 4, it was not until around 700 AD, however, that the Chinese first represented zero in its familiar rounded form as an empty circle. The use of this symbol quickly spread to the Middle East and Europe, where it took on its characteristically oblong shape to avoid confusion with other written characters (Menninger, 1969). The adoption of this digit made it much simpler to use positional systems efficiently.
Traveling Numbers The Chinese character for zero spread rapidly owing to the proliferation of trade between China, the Middle East, and Europe during the Middle Ages. This is an indication that along with the exchange of goods, trade also promoted the development of, useful cognitive technologies. While the intensification of trade surely aided in the spreading of complex numerical systems, it also created the very circumstances in which such ideas would be found useful. Before the establishment of long-distance trade routes, peoples’ main use for numbers was limited to simple, small-scale problems like counting one’s sheep or exchanging small amounts of goods in local markets for one’s family. In such circumstances, there was little incentive to further complicate numerical representations, especially because they require the existence of counterintuitive concepts like the number zero.
Because transporting large quantities of items with pack animals over long distances required complex logistical planning to ensure adequate stocks and supplies, merchants needed to employ new methods for bookkeeping and tracking. As the famous trade routes of the Silk Road (depicted in Figure 5) emerged in the first millennium BC, the distances commodities travelled increased dramatically (Wood, 2002). Competition likely drove merchants to transport larger and larger amounts of goods in their business trips across Eurasia. To accommodate for such changes in the scale of long-distance trade, people probably experimented with new ways to efficiently represent larger quantities. The birth of these economic practices, combined with increased specialization of labor and governmental oversight, gave way to the positional numerical systems used today. Following the lead of the Chinese, societies in the Middle East added the number zero to the Brahmi numerals for 1-9 and created the positional base 10 system used ubiquitously in different cultures today.
The story of numbers is but one of the many examples of the ingenuity that can arise when people borrow and improve upon to the ideas of neighbors. Whether intentional or not, humans learned that by applying written (visual) representations to specific quantities, they were better able to form reasonable conclusions regarding small amounts. When the demands of trade and commerce rendered simple non-positional systems inefficient, the necessary concept of zero was integrated with older systems to eventually produce Hindu-Arabic numerals, mathematics, and other forms of measurement. Without such concerns, however, it really would be more efficient to talk of numbers as Piraha do, and in doing so free up discussion of more pressing concerns.References
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Anthony Stigliani 2010